Abstract:The universal elliptic KZB equation is the integrable connection on the
pro-vector bundle over M_{1,2} whose fiber over the point corresponding to the
elliptic curve E and a non-zero point x of E is the unipotent completion of
\pi_1(E-{0},x). This was written down independently by Calaque, Enriquez and
Etingof (arXiv:math/0702670), and by Levin and Racinet (arXiv:math/0703237). It
generalizes the KZ-equation in genus 0. These notes are in four parts. The
first two parts provide a detailed exposition of this connection (following
Levin-Racinet); the third is a leisurely exploration of the connection in
which, for example, we compute the limit mixed Hodge structure on the unipotent
fundamental group of the Tate curve minus its identity. In the fourth part we
elaborate on ideas of Levin and Racinet and explicitly compute the connection
over the moduli space of elliptic curves with a non-zero abelian differential,
showing that it is defined over Q.